Now You Try

<p>The geometric mean of a set of 11 numbers is the <code class='latex inline'>n</code>th root of the product of the numbers. For example, given two non- consecutive terms of a geometric sequence, 6 and 24, their product is 144 and the geometric mean is <code class='latex inline'>\sqrt{144}</code> or 12. The numbers 6, 12, and 24 form a geometric sequence.</p>
<ul>
<li>Determine the geometric mean of 5 and 125.</li>
</ul>

<p>Which term of the geometric sequence <code class='latex inline'>1, 3, 9</code>, has a value of <code class='latex inline'>19\ 683</code>?</p>

<p>In a geometric sequence, the 3rd term is 20 and the 6th term is -540. Find the first six terms.</p>

<p>Science A certain culture of yeast increases by <code class='latex inline'>\displaystyle 50 \% </code> every three hours. A scientist places 9 grams of the yeast in a culture dish. Write the explicit and recursive formulas for the geometric sequence formed by the growth of the yeast.</p>

<p>A geometric sequence has positive terms. The sum of the first 3 terms of a geometric sequence is 13. The sum of the reciprocals of the first 3 terms is <code class='latex inline'>\frac{13}{9}</code>. Find the first 3 terms of the sequence.</p>

<p>The population of a city increases from 12 000 to 91 125 over 10 years. Determine the annual rate of increase, if the increase
is geometric. </p>

<p>Given the geometric sequence <code class='latex inline'>3, 6, 12, 24, .....</code></p><p>(a) Find the 14th term.</p><p>(b) Which term is 384?</p>

<p> The fourth term in the geometric sequence in part (a) appears as the <code class='latex inline'>n</code>th term in the arithmetic sequence in part (a). Determine the value of <code class='latex inline'>n</code>.</p><p>*<em>Below is part (a) *</em>
(a) Find the arithmetic sequence with first term 1 and common difference not equal to 0, whose second, tenth, and thirty-forth terms are the first 3 terms in a geometric sequence. </p>

<p>Find the common ratio for each geometric sequence.</p><p><code class='latex inline'>\displaystyle 90,-30,10, \ldots </code></p>

<p>Determine the first 4 terms defined by the sequence <code class='latex inline'>t_{1}= 4</code>, <code class='latex inline'>t_{k} = -t_{k - 1} + 2</code>, <code class='latex inline'>k > 1</code>, and determine the general term for this sequence.</p>

<img src="/qimages/56711" />

<p>Find the general term for the following Geometric sequences as well their 10th term.</p><p> 2, 6, 18, 54, 162, ...</p>

<p>Three numbers form an arithmetic sequence with a common difference of <code class='latex inline'>7</code>. When the first term of the sequence is decreased by <code class='latex inline'>3</code>, the second term increased by <code class='latex inline'>7</code>, and the third term doubled, the new numbers form a geometric sequence. What is the original first term? </p><p><strong>(a)</strong> 7</p><p><strong>(b)</strong> 16</p><p><strong>(c)</strong> 20</p><p><strong>(d)</strong> 68</p>

<p>Aryn's dad offer her two different options for her allowance for one year. </p><p>In <strong>option 1</strong>, he would give her <code class='latex inline'>\$25</code> every week. </p><p>In <strong>option 2</strong>, he would give her <code class='latex inline'>\$0.25</code> the first week and then double the amount every following week. </p><p><strong>(a)</strong> Which option represents an arithmetic sequence? Determine the general term for the sequence.</p><p><strong>(b)</strong> Which option represents a geometric sequence? Determine the general term for the sequence.</p><p><strong>(c)</strong> Which plan should her dad pick? Explain.</p>

<p> Find the general term for the following Geometric sequences. </p><p>If <code class='latex inline'>t_{5} = \frac{3}{16}</code> and <code class='latex inline'>t_{8} = \frac{3}{128}</code></p>

<p>This pattern shows the first five steps in constructing the Sierpinski Triangle. Use a pattern to describe the figures.</p><img src="/qimages/87307" />

<p>A chain e-mail starts with one person sending out six e-mail messages. Each of the recipients sends out six messages, and so on. How many e-mail messages will be sent in the sixth round of e-mailing? </p>

<p>Determine if each sequence is arithmetic, geometric, or neither. If it <code class='latex inline'>1</code>'s arithmetic, state the values of <code class='latex inline'>a</code> and <code class='latex inline'>d</code>. If it is geometric, state the values of <code class='latex inline'>a</code> and <code class='latex inline'>r</code>.</p><p><strong>(a)</strong> <code class='latex inline'>x</code>, <code class='latex inline'>3x</code>, <code class='latex inline'>5x</code>, ...</p><p><strong>(b)</strong> 1, <code class='latex inline'>\displaystyle{\frac{x}{2}}</code>, <code class='latex inline'>\displaystyle{\frac{x^2}{4}}</code>, ...</p><p><strong>(c)</strong> <code class='latex inline'>\displaystyle{\frac{m^2}{n}}</code>, <code class='latex inline'>\displaystyle{\frac{m^3}{2n}}</code>, <code class='latex inline'>\displaystyle{\frac{m^4}{3n}}</code>, ...</p><p><strong>(d)</strong> <code class='latex inline'>\displaystyle{\frac{5x}{10}}</code>, <code class='latex inline'>\displaystyle{\frac{5x}{10^3}}</code>, <code class='latex inline'>\displaystyle{\frac{5x}{10^5}}</code>, ...</p>

<p>The first three terms of the sequence -8, <code class='latex inline'>x</code>, <code class='latex inline'>y</code>, <code class='latex inline'>72</code> form an arithmetic sequence, while the second, third, and fourth terms
form a geometric sequence. Determine <code class='latex inline'>x</code> and <code class='latex inline'>y</code>. </p>

<p><code class='latex inline'>Listeria</code> <code class='latex inline'>monocytogenes</code> is a bacteria that rarely causes food poisoning. At a temperature of 10 <code class='latex inline'>^\circ</code>C, it takes about 7 h for the bacteria to double. If the bacteria count in a sample of food is 100, how long will it be until the count exceeds 1 000 000? </p>

<p>A geometric sequence has the property that each term is the sum of the previous two terms. If the first term is <code class='latex inline'>2</code>. what is one possibility for the second term? </p><p><strong>(a)</strong> <code class='latex inline'>-4+\sqrt{3}</code></p><p><strong>(b)</strong> <code class='latex inline'>1-\sqrt{5}</code></p><p><strong>(c)</strong> <code class='latex inline'>4-\sqrt{3}</code></p><p><strong>(d)</strong> <code class='latex inline'>-1+\sqrt{5}</code> </p>

<p>If <code class='latex inline'>t_{n} = 243</code> and <code class='latex inline'>t_{2n - 3} = 6561</code> in a geometric sequence, find the value of <code class='latex inline'>n</code> if <code class='latex inline'>r</code> is a natural number.</p>

<p>A geometric sequence has an initial value of 18 and a common ratio of <code class='latex inline'>\displaystyle \frac{1}{2} </code>. Write a function to represent this sequence. Graph the function.</p>

<p>Find the general term for the following Geometric sequences as well their 10th term.</p><p>32, -16, 8, -4, 2, ... </p>

<p>Critical Thinking Describe the similarities and differences between arithmetic and geometric sequences.</p>

<p>Find the equivalent geometric sequence of <code class='latex inline'>t_{n}</code> of if <code class='latex inline'>t_{1} = 3</code>, and <code class='latex inline'>t_{n + 1} = 2t_{n}</code>.</p>

<p>In an geometric sequence <code class='latex inline'>t_1 + t_2 + t_3 = 21</code>, and <code class='latex inline'>t_{4} + t_{5} + t_{6} = 300</code>. Find <code class='latex inline'>t_n</code>.</p>

<p>Determine all possible values of <code class='latex inline'>x</code> such that <code class='latex inline'>x - 2</code>, <code class='latex inline'>-2 - x</code>, <code class='latex inline'>x + 10</code> are consecutive terms in a geometric sequence.</p>

<p>Cesium-137 (Cs-137), is very dangerous to human life as it accumulates in the soil, the water, and the body. It is believed by scientists that a contamination of Cs-137 of over <code class='latex inline'>1</code> Ci/km<code class='latex inline'>^2</code> (curie per square kilometre) is dangerous. </p><p><strong>(a)</strong> Determine the amount of Cs-137 per square kilometre if about <code class='latex inline'>1.5 \times 10^6</code> Ci of this radioactive substance was released into the environment and spread over an area of about 135 000 km<code class='latex inline'>^2</code>.</p><p><strong>(b)</strong> The half-life of Cs-137 is 30 years. Write an explicit formula to represent the level of Cs-137 left after <code class='latex inline'>n</code> years. How long will it take for the contamination to reach safe levels?</p>

<p>State the common ratio for each geometric sequence and write the next three terms. </p><p><strong>(a)</strong> 1, 2, 4, 8, ...</p><p><strong>(b)</strong> -3, 9, -27, 81, ...</p><p><strong>(c)</strong> <code class='latex inline'>\displaystyle{\frac{2}{3}}</code>, <code class='latex inline'>-\displaystyle{\frac{2}{3}}</code>, <code class='latex inline'>\displaystyle{\frac{2}{3}}</code>, <code class='latex inline'>-\displaystyle{\frac{2}{3}}</code>, ...</p><p><strong>(d)</strong> 600, -300, 150, -75, ...</p><p><strong>(e)</strong> -15, -15, -15, -15, ...</p><p><strong>(f)</strong> 0.3, 3, 30, 300, ...</p><p><strong>(g)</strong> 72, 36, 18, 9, ...</p><p><strong>(h)</strong> <code class='latex inline'>x</code>, <code class='latex inline'>x^3</code>, <code class='latex inline'>x^5</code>, <code class='latex inline'>x^7</code>, ... </p>

<p>Determine the number of terms in each geometric sequence. </p><p><strong>(a)</strong> 6, 18, 54, ...,4374</p><p><strong>(b)</strong> <code class='latex inline'>0.1, 100, 100 000, ..., 10^{14}</code></p><p><strong>(c)</strong> <code class='latex inline'>5, -10, 20, ..., -10 240</code></p><p><strong>(d)</strong> <code class='latex inline'>3, 3\sqrt{3}, 9, ..., 177 147</code></p><p><strong>(e)</strong> <code class='latex inline'>31 250, 6250, 1250, ..., 0.4</code></p><p><strong>(f)</strong> <code class='latex inline'>16, -8, 4, ..., \displaystyle{\frac{1}{4}}</code></p>

<img src="/qimages/56708" />

<p>The differences between consecutive terms in a geometric sequence form a new geometric sequence. For instance, when you take the differences between the consecutive terms of the geometric sequence <code class='latex inline'>\displaystyle 5,15,45,135, \ldots </code> you get <code class='latex inline'>\displaystyle 15-5,45-15,135-45, \ldots </code> The new geometric sequence is <code class='latex inline'>\displaystyle 10,30,90, \ldots </code> Compare the two sequences. How are they similar, and how do they differ?</p>

<p>Identify a pattern and find the next three numbers in the pattern.
<code class='latex inline'>\displaystyle 6,12,18,24, \ldots </code></p>

<p>The geometric mean of a set of 11 numbers is the <code class='latex inline'>n</code>th root of the product of the numbers. For example, given two non- consecutive terms of a geometric sequence, 6 and 24, their product is 144 and the geometric mean is <code class='latex inline'>\sqrt{144}</code> or 12. The numbers 6, 12, and 24 form a geometric sequence.</p>
<ul>
<li>Insert three geometric means between 4 and 324.</li>
</ul>

<p>Determine Whether the sequence is arithmetic, geometric, or neither. Give a reason for your answer. </p><p><strong>(a)</strong> <code class='latex inline'>5, 3, 1, -1, ...</code></p><p><strong>(b)</strong> <code class='latex inline'>5, -10, 20, -40, ...</code></p><p><strong>(c)</strong> 4, 0.4, 0.04, 0.004, ...</p><p><strong>(d)</strong> <code class='latex inline'>\displaystyle{\frac{1}{2}}</code>, <code class='latex inline'>\displaystyle{\frac{1}{6}}</code>, <code class='latex inline'>\displaystyle{\frac{1}{18}}</code>, <code class='latex inline'>\displaystyle{\frac{1}{54}}</code>, ...</p><p><strong>(e)</strong> 1, <code class='latex inline'>\sqrt{2}</code>, <code class='latex inline'>\sqrt{3}</code>, 2, <code class='latex inline'>\sqrt{5}</code>, ...</p><p><strong>(f)</strong> 1, 5, 2, 5, ...</p>

<p>Film speed is the measure of a photographic film’s sensitivity to light. The ISO (International Organization of Standardization) film-speed scale forms a geometric sequence. If the first term in the sequence is <code class='latex inline'>25</code> and the fourth term is <code class='latex inline'>50</code>, what is the fifth term?</p>

<p>Error Analysis A friend says that the</p><p>recursive formula for the geometric sequence <code class='latex inline'>\displaystyle 1,-1,1,-1,1, \ldots </code> is <code class='latex inline'>\displaystyle a_{n}=1 \cdot(-1)^{n-1} . </code> Explain your friend's error and give the correct recursive formula for the sequence.</p>

<p>Think about a Plan Suppose you are rehearsing for a concert. You plan to rehearse the piece you will perform four times the first day and then to double the number of times you rehearse the piece each day until the concert. What are two formulas you can write to describe the sequence of how many times you will rehearse the piece each day?</p>
<ul>
<li><p>How can you write a sequence of numbers to represent this situation?</p></li>
<li><p>Is the sequence arithmetic, geometric, or neither?</p></li>
<li><p>How can you write explicit and recursive formulas for this sequence?</p></li>
</ul>

<p>If 4, a, b, c, 324 form a geometric sequence, find a, b, c.</p>

<p>A store manager plans to offer discounts on some sweaters according to this sequence:</p><p><code class='latex inline'>\displaystyle \$ 48, \$ 36, \$ 27, \$ 20.25, \ldots </code> Write the explicit and recursive formulas for the sequence.</p>

<p>Find the general term for the following Geometric sequences as well their 10th term.</p><p><code class='latex inline'>2/5, \frac{4}{15}, \frac{8}{45}, ...</code></p>

<p> Find the general term for the following Geometric sequences. </p><p>If <code class='latex inline'>t_{26} = -1</code> and <code class='latex inline'>t_{27} = -49</code></p>

<p>Which term of the geometric sequence <code class='latex inline'>\displaystyle{\frac{3}{64}}</code>, <code class='latex inline'>-\displaystyle{\frac{3}{16}}</code>, <code class='latex inline'>\displaystyle{\frac{3}{4}}</code>, ... has a value of <code class='latex inline'>192</code>?</p>

<p>In a certain country, elections are held every <code class='latex inline'>4</code> years. Voter turnout at elections increases by <code class='latex inline'>2.6\%</code> each time an election is held. In 1850, when the country was formed, 1 million people voted.</p><p><strong>(a)</strong> Determine an equation to model the number of voters at any election. Graph the equation.</p><p><strong>(b)</strong> Is this function continuous or discrete? Explain your answer.</p><p><strong>(c)</strong> How many people vote in the 2010 election?</p>

<p>Refer to question 14. Determine three geometric means between <code class='latex inline'>x^5+x^4</code> and <code class='latex inline'>x+1</code>. </p><p>(Question 14)
The geometric mean of a set of 11 numbers is the <code class='latex inline'>n</code>th root of the product of the numbers. For example, given two non- consecutive terms of a geometric sequence, 6 and 24, their product is 144 and the geometric mean is <code class='latex inline'>\sqrt{144}</code> or 12. The numbers 6, 12, and 24 form a geometric sequence.</p><p><strong>(a)</strong> Determine the geometric mean of 5 and 125.</p><p><strong>(b)</strong> Insert three geometric means between 4 and 324.</p>

<p>Determine the value(s) of <code class='latex inline'>y</code> if <code class='latex inline'>4y+1</code>, <code class='latex inline'>y+4</code>, and <code class='latex inline'>10-y</code> are consecutive terms in a geometric sequence. </p>

<p>Find the common ratio for each geometric sequence.</p><p><code class='latex inline'>\displaystyle -9,4.5,-2.25, ... </code></p>

<p> Find the general term for the following Geometric sequences. </p><p>If <code class='latex inline'>t_{4} = 24</code> and <code class='latex inline'>t_{6} = 54</code></p>

<p>The arithmetic mean of two numbers is 65. The geometric mean of the same two number is 25. Find the numbers.</p>

<p>Find the arithmetic sequence with first term 1 and common difference not equal to 0, whose second, tenth, and thirty-forth terms are the first 3 terms in a geometric sequence. </p>

<p>Determine <code class='latex inline'>x</code> and <code class='latex inline'>y</code> for each geometric sequence. </p><p><code class='latex inline'> 3, x, 12, y, ...</code></p>

<p>Write the first four terms of each geometric sequence. </p><p><strong>(a)</strong> <code class='latex inline'>t_n=5(2)^{n-1}</code></p><p><strong>(b)</strong> <code class='latex inline'>a=500</code>, <code class='latex inline'>r=-5</code></p><p><strong>(c)</strong> <code class='latex inline'>f(n)=\displaystyle{\frac{1}{4}(-3)^{n-1}}</code></p><p><strong>(d)</strong> <code class='latex inline'>f(n)=2(\sqrt{2})^{n-1}</code></p><p><strong>(e)</strong> <code class='latex inline'>a=-1</code>, <code class='latex inline'>r=\displaystyle{\frac{1}{5}}</code></p><p><strong>(f)</strong> <code class='latex inline'>t_n=-100(-0.2)^{n-1}</code></p>

<p>Open-Ended Write a geometric sequence. Then write the explicit and recursive formulas for your sequence.</p>

<p>all but the middle square are shaded. This process is repeated with the remaining shaded squares to produce a fractal called the Sierpinski carpet. </p><img src="/qimages/515" /><p><strong>(a)</strong> Use grid paper to produce the first five stages of the fractal.</p><p><strong>(b)</strong> Write a formula to determine the shaded area at each stage.</p><p><strong>(c)</strong> Use the formula to determine the shaded area at stage 20.</p><p><strong>(d)</strong> Research this fractal. When was it first explored?</p>

<p>If <code class='latex inline'>4, x, 324</code> form a geometric sequence, find <code class='latex inline'>x</code>.</p>

<p>Determine <code class='latex inline'>x</code> and <code class='latex inline'>y</code> for each geometric sequence. </p><p><code class='latex inline'>-2, x, y, 1024, ...</code></p>

<p>For each geometric sequence, determine the formula for the general term and then write <code class='latex inline'>t_{9}</code>. </p><p><strong>(a)</strong> 54, 18, 6, ...</p><p><strong>(b)</strong> 4, 20, 100, ...</p><p><strong>(c)</strong> <code class='latex inline'>\displaystyle{\frac{1}{6}}</code>, <code class='latex inline'>\displaystyle{\frac{1}{5}}</code>, <code class='latex inline'>\displaystyle{\frac{6}{25}}</code>, ...</p><p><strong>(d)</strong> 0.0025, 0.025, 0.25, ...</p>